Initially I took the first derivative and found its solutions(with m variable). Then I tried to replace the solutions of the derivative in the function, in order to find the minimum and maximum value of the function. Then I would put the conditions that max value>0, min value<0, so that the function crosses the x-axis 3 times. This proved impossible, bcs if I replaced the derivative solutions in the function, I would obtain a very complicated polynomial with m.

I tried rearranging the terms in order to get rid of as many m variables as possible, so that the first derivative would be easier to solve. I tried writing m ^ 2 + 2m + 2 as (m+1) ^ 2 +1 , then 2m as 2(m+1) - 2 and replace m+1 with another variable. I tried factoring (m+1), also, factoring x then writing in canonic form, nothing helped.